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I’ve been getting more into theoretical/analytical modeling lately, and I’ve been playing around with software to help me do some of the more complicated calculus involved (for two reasons: 1) I’m lazy, 2) My skills are very rusty). Python (of course) provides excellent symbolic capabilities through the Sympy module. I played around with it to try and get a feel for how it works, but I couldn’t find any help online, nor anyone who has posted a tutorial, on analyzing basic biological/ecological models with Sympy. So here is my version. Below, I solve both the exponential growth and logistic growth models using Sympy, then plot the results. Here is a step-by-step tutorial for Bio-Sympy.

Exponential Growth

Exponential growth is defined by the differential equation

\( \frac{dy(t)}{dt} = k*y(t) \)

and this ODE has the analytical solution of

\( y(t) = Y_0 e^{kt} \)

So how do we use Sympy to go from the ODE to the general solution? Well, like this.

First, gotta import the modules we need, and then initialize pretty printing just so the output is readable (see the Sympy docs for this):

Next, define the variables k (the intrinsic growth rate) and t (for time). Then, make y a function of t

from sympy.abc import t, k
y = sm.Function('y')(t)

Then, define the derivative of y with respect to t (the left-hand side of the ODE), and then define the right-hand side of the ODE.

dy = y.diff(t)
rhs = k*y

We need to set these two quantities equal to one another, as in the ODE above, which we can do using a Sympy Equality

eq = sm.Eq(dy, rhs)

If you print eq, it should give you the differential equation. Now that we have the differential equation set up, all we need to do is solve. Since this is a simple one, Sympy can do it on its own with no hints or guesses:

sol = sm.dsolve(eq)
sol

Where sol should give you the analytical solution \( y(t) = C_1 e^{kt} \). But \( C_1 \) here is just a constant, we want to put it in terms of the initial conditions. (I know this is trivial in this case, but bear with me). First, we need to find the initial conditions. We do so by substituting 0 in for t, and then setting that equal to n0.

# find out what C is in terms of y0
t0 = sol.args[1].subs({'t':0})
n0 = sm.symbols('n0')
eq_init = sm.Eq(n0, t0)
# it takes a little more work to isolate C1 here
# try each step for yourself to see what it does
C1 = t0.args[2].args[0].args[0]
t0_sol = sm.solve(eq_init, C1)

The initial conditions are also rather complex:

\( C_1 =\frac{\log \left ( \frac{-n_0)}{K-n_0} \right )}{K} \)

But we can substitute that back into the original solution to get the general solution in terms of the initial conditions

# substitute the expression for C1 back into the equation
final = sol.args[1].subs(C1, t0_sol[0])

In my last post, I discussed how to get predictions from Gaussian Processes in STAN quickly using the analytical solution. I was able to get it down to 3.8 seconds, which is pretty quick. But I can do better.

One of this issues you may or may not have noticed is, using the model wherein posterior predictions are generated quantities, your computer bogs down with anything over 1000 iterations (I know mine froze up). The issue here is that generated quantities saves all the variables into memory, including those large matrices \( \boldsymbol{K}_{obs}, \boldsymbol{K}_{obs}^{*}, \boldsymbol{K}^{*} \) . We can resolve this issue, and speed things up, using functions to calculate the predictive values while storing the matrices only as local variables within the function (they are not saved in memory).

One obvious solution is to not calculate them within STAN, but to do it externally. Python can do this relatively quickly with judicious use of numpy (whose linear algebra functions are all written in C and so very fast) and numba/just-in-time compilers. However, since STAN compiles everything into C++, and many of its linear algebra functions are optimized for speed, it makes sense to do it all in STAN. Fortunately, we can create our own functions within STAN to do this, which are then compiled and executed in C++. The code is as follows:

There are a couple of tricks here. I used a diagonal matrix \(\boldsymbol{I}\sigma_n\) to remove a for-loop in calculating the noise variance. It has the added benefit of cleaning up the code (I find for-loops to be messy to read, I like things tidy). Second, by using functions, the generated quantities only stores the predictions, not all the extra matrices, freeing up tons of memory.

One other thing to note is that the prediction function ends with the suffix ‘_rng’. This is because it is a random number generator, drawing random observations from the posterior distribution. Using the ‘_rng’ suffix allows the function to access other ‘_rng’ functions, chiefly the normal_rng function which draws random normal deviates. That’s necessary for the cholesky trick to turn N(0,1) numbers into the posterior distribution (see the last post). However, normal_rng only returns ONE number, so you have to repeat it for however many observations you need, hence the rep_vector( ) wrapper.

This code is extremely fast. After compilation, it executes 4 chains, 1000 iterations in roughly 1 second (compared to just over three from the previous post). Further, since I no longer have memory issues, I can run more iterations. Whereas 5000 iterations froze my computer before, now it executes in 5 seconds. 10,000 iterations, previously unimaginable on my laptop, runs in 10 seconds.

As described in an earlier post, Gaussian process models are a fast, flexible tool for making predictions. They’re relatively easy to program if you happen to know the parameters of your covariance function/kernel, but what if you want to estimate them from the data? There are several methods available, but my favorite so far is STAN. True, it requires programming the kernel by hand, but I actually find this easier to understand than trying to parse out the kernel functions from, say, scikit-learn.

STAN can fit GP models quickly, but there are certain tricks you can do that make it lightning fast and accurate. I’ve had trouble getting scikit to converge on a stable/accurate solution, but STAN does this with no problem. Plus, the Hamiltonian Monte-Carlo sampler is very quick for GPs (see the STAN User Manual for more).

Here’s a quick tutorial on how to fit GPs in STAN, and how to speed them up. First, let’s import our modules and simulate some fake data:

This model fits the parameters of the kernel. On my computer, it does so in about 0.15 seconds total (1000 iterations).

Now that we know what the hyperparameters are, we’d like to predict new values, but we also want to do so incorporating full uncertainty in the hyperparameters. The way the STAN manual says to go about this is to make a second vector containing the locations you’d like to predict, paste those together to the X values you have, make a vector of prediction points as parameters, paste those to the Y values you have, and feed those into the multivariate normal distribution as one big mush. The issue here is that the covariance matrix, Sigma, gets very large very fast. Large covariance matrices take a while to invert in the multivariate probability density, and so slows down the sampler.

Here’s the model, after making a vector of 100 prediction points to get a smooth line:

This is extremely slow, taking about 81 seconds on my computer (up from less than one second). Taking the cholesky decompose of Sigma an using multi_normal_cholesky didn’t speed things up, either.

We can speed this up by taking advantage of the analytical form of the solution. That is, once we know the hyperparameters of the kernel from the observed data, we can directly calculate the multivariate normal distribution of the predicted data:

We can calculate those quantities directly within STAN. Then, as an added trick, we can take advantage of the Cholesky decomposition to generate random samples of \( Y_p \) within STAN as well. Here’s the annotated model:

Gaussian Processes for Machine Learning by Rasmussen and Williams has become the quintessential book for learning Gaussian Processes. They kindly provide their own software that runs in MATLAB or Octave in order to run GPs. However, I find it easiest to learn by programming on my own, and my language of choice is Python. This is the first in a series of posts that will go over GPs in Python and how to produce the figures, graphs, and results presented in Rasmussen and Williams.

Figure 2.2

We want to make smooth lines to start, so make 100 evenly spaced \(x\) values:

N_star = 101
x_star = np.linspace(-5, 5, N_star)

Next we have to calculate the covariances between all the observations and store them in the matrix \(\boldsymbol{K}\). Here, we use the squared exponential covariance: \(\text{exp}[-\frac{1}{2}(x_i – x_j)^2]\)

We now need to calculate the covariance between our unobserved data (x_star) and our observed data (x_obs), as well as the covariance among x_obs points as well. The first for loop calculates observed covariances. The second for loop calculates observed-new covariances.

This may not look exactly like the Rasmussen and Williams Fig. 2.2b because I guessed at the data points and they may not be quite right. As the authors point out, we can actually plot what the covariance looks like for difference x-values, say \(x=-1,2,3\).

In this case, however, we’ve forced the scale to be equal to 1, that is you have to be at least one unit away on the x-axis before you begin to see large changes \(y\). We can incorporate a scale parameter \(\lambda\) to change that. We can use another parameter \(\sigma_f^2\) to control the noise in the signal (that is, how close to the points does the line have to pass) and we can add further noise by assuming measurement error \(\sigma_n^2\).